
where \(z_{k}\sim N(0,I)\)  is an independent standard Gaussian random
variable in \(\mathbb {R}^{d}\) . As \(x_{k}\)  is updated following ULA,
let \(p_{k}\)  represent its probability distribution. It is known that
ULA converges to a biased limiting distribution \(\pi _{\eta }\ne \pi \) 
for any fixed \(\eta >0\) . Thus, \(H(p_{k}|\pi )\)  does not converge to
0 along ULA, as it has an asymptotic bias \(H(\pi _{\eta }|\pi )>0\) .
When the true target distribution \(\pi \)  complies with an \(\alpha \) -mixture
weakly smoothness and LSI, we can prove convergence in KL divergence
along ULA. A key observation is that ULA algorithm will converge uniformly
in time if the discretization error between the ULA output in one
iteration and the Langevin dynamics is bounded. This technique has
been used in many papers, [1]}, [2]}.
Our proof structure is similar to that of [1]},
whose analysis needs stronger assumptions.
